Chandrasekhar’s limit
As a part of my graduate school coursework, I had a chance to go through the works of Dr. Subrahmanyan Chandrasekhar, renowned astrophysicist, and Nobel laureate. There are numerous articles extolling his contribution to physics, and chronicling his growth as a child prodigy. However, I wanted to pay a homage, however small, by attempting to explain one of his most seminal contributions to astrophysics: the existence of the “Chandrasekhar limit”.
Chandrasekhar limit is not famous because of an inherent mystique nature of the limit — rather, it is famous because it brought together multiple branches of physics, and theoretically predicted a phenomenon yet to be observed. He predicted the existence of an upper bound on the mass of the core of a star, for it to remain a White dwarf. Any heavier, the core is doomed to collapse due to its self-gravity!
To understand the limit, basic branches of science are brought together: Quantum mechanics, Special relativity, Newtonian gravity and Fluid mechanics. So let’s jump right in!
The first sections here will be fully qualitative. A primer to the calculations with references will be given at the end for people interested in working out the mathematics!
Qualitative Explanation
Macroscopic physics: Fluid mechanics, Newtonian gravity
While fluid mechanics indeed arises from microscopic interactions between particles, considering we have at least 10²³ particles (that’s small, by the way; a teaspoon of table salt probably contains as many particles), we can consider a fluid system to be continuous, and forget (for now!) about it’s discrete nature. Stars, we know, are large balls of fluid. Thus, we can use a continuum approach to write down certain equations which can help us determine properties of stars.
Stars are also massive fluid systems — so what supports them from collapsing under their self-gravity? It must be some sort of pressure, which ‘pushes’ the star apart and resist any collapse. To a first order, if we ignore radiation effects and plasma effects, we can talk about the condition of a star being in Hydrostatic equilibrium.
Hydrostatic equilibrium is the condition of absence of any sort of acceleration in a fluid system. One could as well say there are no external forces on the system, and of course there are no viscous forces on the system.
For such a system in hydrostatic equilibrium, it is obvious that the pressure force from inside must match the self gravity trying to collapse the star. Thus, we get one equation relating pressure and the gravitational potential from this force balance condition. This equation is called the Momentum equation.
Now, what determines how much gravity we have? This is given by a theorem of Newton:
The total force exerted by a sphere of mass ‘M’ at a another mass ‘m’ at a distance ‘r’ from the center of the sphere is equivalent to the force exerted by a point at the center of the sphere and having the same mass ‘M’.
This equation is given as the Poisson equation for gravitational potential. Thus, we have two equations which relate: (i). Density, (ii). Pressure and (iii). Gravitational potential. As it is, we have 2 equations and 3 variables, and thus we need one more equation to actually describe the star. An equation, which would relate pressure and density (since we have pressure-potential and potential-density relations) is needed, an equation known as The Equation of state.
And this is where Quantum Mechanics (and Special Relativity, later) comes into the picture.
Microphysics: Quantum Mechanics
The historical argument goes as such: a given star supports itself against its own gravity by fusing together Hydrogen to produce Helium and producing energy. But, there must come a time when the Hydrogen must be fully converted to Helium (theoretically speaking), and thus further collapse must occur.
Now of course, Helium can be fused, and it is indeed what happens — the fusion of Helium, Carbon and so on proceeds till Iron in certain stars, after which any further fusion was not possible — thus, the star would collapse for eternity. Earlier, people did not know what would happen at such pressures, and it seemed the stellar structure theory is going for a ride — at least till William Fowler ‘saved’ it.
Fowler’s argument can be summarized elegantly using one of the fundamental principles of quantum mechanics — Pauli exclusion principle. This principle applies to particles called Fermions, which are essentially particles like electrons, protons, neutrons, etc. The principle goes as follows:
Two or more identical Fermions cannot occupy the same quantum mechanical state simultaneously.
That is, if we have two electrons, and we are given a level ‘E’, the principle states these two electrons cannot occupy the same level — one of the electrons will be in this level, and the other electron must be in some other level. Of course, electrons have something called ‘Spin’, which must be taken into account while performing this principle. The principle has been summarized in Fig. 1.
Fowler, striking a brilliant argument, said since no two electrons can occupy the same quantum mechanical state, at the large pressures in the cores of a fully ‘burnt’ stars, one must necessarily have a pressure due to this exclusion principle supporting gravity. And if we consider the whole of the core (which is what remains at the end — the outer layers are all gone by now!), it must be supported by the degenerate electron gas. The electrons are called degenerate, because they fill up all of the levels from the lowest energy level to a peak level, called the ‘Fermi energy level’. That is, if one plots the number of electrons in a given energy level, the plot looks like how it is shown in Fig.2.
Fermi had thus saved stars from eternal collapse. However, Chandrasekhar came along, and had a look at this phenomenon. Now, he used another pillar of Quantum mechanics — the Uncertainty principle along with Special relativity. The principle, due to Heisenberg can be interpreted as follows:
The position and momentum cannot be measured to arbitrary accuracy simultaneously. If the position of a particle is known precisely, its momentum information will be imprecise and vice-versa.
Chandrasekhar, from a theoretical formulation, built upon Fowler’s work. Consider the electrons initially supporting the stellar core by degeneracy pressure. Now, if the mass were large, the volume of the core would be smaller (since gravity will try to crumple the core). As the volume is low, the electrons’ positions in the core are made more precise, thus their momenta will have higher uncertainties. Roughly speaking, at one particular mass of the core, it will so happen that these electrons will have high momentum uncertainty — high enough to make them relativistic. Chandrasekhar worked out the general form of pressure variation with density, and took the relativistic limit. And to his surprise, the pressure increase with density offered by the degenerate electrons drops as they become relativistic!
Thus, the pressure initially was high enough to sustain gravity. At a higher mass, due to relativistic effects, the quantum pressure due to the electrons is unable to support the star against gravity — thereby destroying Fowler’s save. This mass limit wherein the relativistic degenerate electron gas pressure just supports gravity, is called the Chandrasekhar limit. It’s value is around 1.4 times the mass of our Sun. Any core, with the mass more than this limit, will resume its collapse!
What happens to a core of more than 1.4 Solar mass? Well, crudely speaking, the pressure is great enough to fuse electrons and protons to create neutrons, which then provide degeneracy pressure at some stage. These stars are called Neutron stars (this is not exactly correct, and I am not well-versed in this topic to go into detail, since the densities are so high, Newtonian physics goes for a ride. General relativity comes into the picture!). Collapse past the Neutron star limit leads to the formation of Black holes, about which I have no idea!
Also, one must ask the question — why are electrons going relativistic first? What about protons? Well, electrons are much lighter than protons, so the increase in momentum results in much higher velocities for electrons rather than protons!
It is to be appreciated that Chandrasekhar derived this limit entirely from theory. He presented an exact result, the relativistic limit of which gives the mass limit. Thus, in the 1930s, he combined fluid mechanics, quantum mechanics and special relativity to predict the presence of a maximum limit of a star supported by electron degeneracy pressure. This later became a bone of contention between him and Arthur Eddington (since Eddington, being an observer, found it absurd for a star to keep collapsing for eternity!), though we shall not dwell into it!
Well, I did promise a primer in the mathematics that goes behind the derivation of this limit. The derivation present here follows the class lectures of Prof. Kandaswamy Subramanian in Statistical Mechanics for the IUCAA-NCRA graduate school 2018.
Quantitative explanation
The fluid and gravity equations are easy enough to understand, if not derive. Basically, one considers a spherically symmetric system (i.e, no dependence of any variable on the spherical angles, and the dependence is only on the radius), and gradients existing only in the radial direction. In such a case, the momentum equation takes the form:
The Poisson equation for gravity, due to spherical symmetry simplification, takes the form:
Now, we just need to find the equation of state. For this case, we must go to Fermi-Dirac statistics.
Fermi-Dirac statistics presents the distribution of Fermions described earlier — precisely speaking, it gives a distribution function for the distribution of N Fermions among K levels, where no two Fermions can occupy the same level. The derivation is involved (to do it properly, one must start with the density of states), and the interested reader is pointed to a standard source (at the end). However, all we need to take here is the curve for occupation of states at T=0K, present in Fig.2. For such a system, one can define the number density as:
We can similarly define the energy for the system of electrons. Thus, the energy of the system can be given as:
Taking the two limits, one can obtain the dependence of energy E on the fermi momentum p_F, which can in turn be expressed in terms of the number density n (using the equation shown in Fig.5). Thus, we can obtain Energy in terms of the number density. We are now one step closer to getting an Equation of state.
Now, from Thermodynamics, we know:
The pressure can be expressed in terms of energy for a constant entropy as:
Thus, we can obtain the pressure in terms of volume ( or rather, the density ) by using an effective mass and thus get our equation of state as:
Thus, we now have our equation of state. The constants of proportionality contain factors of the Planck’s constant, and mean molecular mass, obtained on solving the energy and number density correctly. One can make substitutions from the momentum equation and the equation of state to the Poisson equation, and obtain (upon redefining the power of density in the equation of state as 1+1/n, where n is an integer) what is known as the Lane-Emden equation:
Indeed, I am just outlining the steps and not deriving the result step-by-step. ‘f’ is a measure of the potential without any dimensions. ‘x’ is the non-dimensional length variable, and runs from 0 to 1. The boundary conditions used to solve this equation are (i). f(1)=0, and (ii). f’(0) = 0, where f’(x) denotes the derivative. The former condition defines the boundary of the star, and the latter condition puts a constant density at one approaches the center of the star (since a star does not have infinite mass!).
The equation need not be solved fully, since we just need the mass of the star for which the relativistic degeneracy pressure balances gravity. This is simple:
Thus, numerically, we can obtain this gradient! Just taking care of the other factors, and using the mean molecular mass of Helium (considering fully-ionized Helium, since it forms the major component of a White Dwarf, and the reactions do not really proceed all the way to Iron) as 2, we obtain:
That’s it! There are plenty of places to ask questions in this derivation here, since I have not gone through each step. However, for those interested, I have uploaded the code to solve the Lane-Emden equation on Git — it can be run directly on Google Colab: https://github.com/Vishal-Upendran/Astro101/blob/master/FluidMechanics_SukantaBose/PolytropicEOS.ipynb
That presents a very, very brief overview of Chandrasekhar’s limit. There are many more things to it, though:
- Why do we consider the whole star to be uniformly degenerate? Can it be degenerate only till a certain radius, and above it there is still some burning happening?
- Will very massive stars always collapse and by-pass the Chandrasekhar limit? Will they emit any material out which finally results in a core of less than 1.4 Solar mass?
There are some interesting ways to look at White dwarfs, and degenerate matter. References are provided below for the brave reader!
References
1. Pauli exclusion principle image from Chem libre texts: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Electronic_Structure_of_Atoms_and_Molecules/Electronic_Configurations/Pauli_Exclusion_Principle
2. Fermi energy level diagram: https://universe-review.ca/R08-04-degeneracy.htm
3. The lecture notes of Prof. Kandaswamy are not available yet — I shall ask him for sharing on the internet. However, the derivations may be found in:
3.1: Landau and Lifshitz, Statistical Physics, “Equilibrium of bodies of large mass” — this has the derivation similar to how I have done here. It also explains the derivation of distribution function of Fermions.
3.2: S. Chandrasekhar, “An Introduction to the study of Stellar Structure” — this is THE book to read. Chandrasekhar’s writing style is very different, and it takes time getting used to — but it is lucid. It contains the equations in their full glory and Chandrasekhar works out through the theory step-by-step. It is a must read if one must understand the theory of stellar structure. It is my preferred book of choice.
4. G. Venkataraman, “Chandrasekhar and his limit” — a light read, with much more detail than this blog post. There mathematics involved, but it is not a textbook, but rather meant to engage a beginner into the beauty of Chandra’s theory.
5. All the texed images of equations were done using http://latex2png.com/. Amazing place to get high-res images of LaTeX equations!
